For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Łojasiewicz inequalities. We also discuss the more general case of λ-convex functions and we provide a general convergence theorem for the corresponding gradient dynamics. Specialising our results to the Boltzmann entropy, we recover Otto-Villani's theorem asserting the equivalence between logarithmic Sobolev and Talagrand's inequalities. The choice of power-type entropies shows a new equivalence between Gagliardo-Nirenberg inequality and a nonlinear Talagrand inequality. Some nonconvex results and other types of equivalences are discussed.
Lojasiewicz inequality; Functional inequalities; Gradient flows; Optimal Transport; Monge-Kantorovich distance;
Adrien Blanchet, and Jérôme Bolte, “A family of functional inequalities: lojasiewicz inequalities and displacement convex functions”, TSE Working Paper, n. 17-787, March 2017.
Adrien Blanchet, and Jérôme Bolte, “A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions”, Journal of Functional Analysis, vol. 25, n. 7, October 2018, pp. 1650–1673.
Journal of Functional Analysis, vol. 25, n. 7, October 2018, pp. 1650–1673